Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $n \neq 0$. $x = \dfrac{n}{18n - 81} \div \dfrac{7n}{7(2n - 9)} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{n}{18n - 81} \times \dfrac{7(2n - 9)}{7n} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ n \times 7(2n - 9) } { (18n - 81) \times 7n } $ $ x = \dfrac {n \times 7(2n - 9)} {7n \times 9(2n - 9)} $ $ x = \dfrac{7n(2n - 9)}{63n(2n - 9)} $ We can cancel the $2n - 9$ so long as $2n - 9 \neq 0$ Therefore $n \neq \dfrac{9}{2}$ $x = \dfrac{7n \cancel{(2n - 9})}{63n \cancel{(2n - 9)}} = \dfrac{7n}{63n} = \dfrac{1}{9} $